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2002 Almost Sure Stability of Linear Itô-Volterra Equations with Damped Stochastic Perturbations
John Appleby
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Electron. Commun. Probab. 7: 223-234 (2002). DOI: 10.1214/ECP.v7-1063


In this paper we study the a.s. convergence of all solutions of the Itô-Volterra equation \[ dX(t) = (AX(t) + \int_{0}^{t} K(t-s)X(s),ds)\,dt + \Sigma(t)\,dW(t) \] to zero. $A$ is a constant $d\times d$ matrix, $K$ is a $d\times d$ continuous and integrable matrix function, $\Sigma$ is a continuous $d\times r$ matrix function, and $W$ is an $r$-dimensional Brownian motion. We show that when \[ x'(t) = Ax(t) + \int_{0}^{t} K(t-s)x(s)\,ds \] has a uniformly asymptotically stable zero solution, and the resolvent has a polynomial upper bound, then $X$ converges to 0 with probability 1, provided \[ \lim_{t \rightarrow \infty} |\Sigma(t)|^{2}\log t= 0. \] A converse result under a monotonicity restriction on $|\Sigma|$ establishes that the rate of decay for $|\Sigma|$ above is necessary. Equations with bounded delay and neutral equations are also considered.


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John Appleby. "Almost Sure Stability of Linear Itô-Volterra Equations with Damped Stochastic Perturbations." Electron. Commun. Probab. 7 223 - 234, 2002.


Accepted: 21 August 2002; Published: 2002
First available in Project Euclid: 16 May 2016

zbMATH: 1027.60067
MathSciNet: MR1952184
Digital Object Identifier: 10.1214/ECP.v7-1063

Primary: 34K20, 60H10, 60H20
Secondary: 45D05


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